Effect of Secondary Bending and Internal Compressions on the Life of Stranded Wire Ropes with Hemp Core

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Title:

NATIONAL RESEARCH COUNCIL

OF

CANADA Technical Translation TT-208

Effect of secondary bending and internal compressions on the life of stranded wire ropes with hemp core, (Einfluss der

sekundBren Biegung und der inneren Pressungen auf die Lebensdauer von Stahlchaht-Litzenseilen mit Hanf seele, )

Professor Dr, Tho Vtyss,

Swiss Federal Materials Testing Institute,

(EMPA),

ZBrich,

Reference : Schweizerf sche Bauzeitung,

-

67, pp, 193-8, 212-5 and 225-8,

The s t r e s s e s due t o s e c o n d a r y b e n d i n g and t h e com-

p r e s s i o n s between w i r e r o p e and s h e a v e and where two s t r a n d s come i n t o c o n t a c t a r e t h e b a s i c d e t e r m i n i n g f a c t o r s f o r

t h e l o a d and l i f e of s t r a n d e d w i r e r o p e s w i t h hemp c o r e . Formulae f o r c a l c u l a t i n g t h e s e s t r e s s e s and c o m p r e s s i o n s a r e d e r i v e d and d a t a a r e g i v e n on t h e i r a d m i s s i b l e mag- n i t u d e @ I n a d d i t i o n t o t h i s , t h e p l a s t i c d e f o r m a t i o n i s i n v e s t i g a t e d , I t i s shown how t h e formulae c a n be a p p l i e d t o e l e v a t o r r o p e s , t y p e B e However, t h e f o r m u l a e may a l s o be a p p l i e d q u i t e g e n e r a l l y t o c a b l e s f o r mountain r a i l w a y s and t o t r a c t i o n c a b l e s ( d r a g l i n e s ) o f c a b l e r a i l w a y s , s p a n r o p e s , r o p e s i n mines, e t c , The p r e s e n t t r e a t i s e i s p a r t o f a more comprehensive work e n t i t l e d 'Untersuchungen Uber d i e Beanspruchungen und Berechnung von S t a h l d r a h t s e i l e n d e r Schwebe- und S t a n d s e i l b a h n e n " ( I n v e s t i g a t i o n s r e g a r d i n g t h e l o a d and c a l c u l a t i o n o f c a b l e s of suspended r a i l w a y s and c a b l e w a y s ) which w i l l be p u b l i s h e d a s EMPA r e p o r t No, 1 6 6 i n t h e n e a r f u t u r e .

A , S t r e s s e s R e s u l t i n g from Secondary Bending

I n s t e e l w i r e r o p e s s e c o n d a r y b e n d i n g o c c u r s a t a l l p o i n t s where t h e i n d i v i d u a l l a y e r s of w i r e i n t e r s e c t and t h e t r a n s v e r s e f o r c e Po a c t i n g on a n o u t e r w i r e ( F i g , 1 ) makes i t s e l f f e l t , e , g o a t t h e c e n t r e o f s u c h a f i e l d ,

f o l l o w s : d = d + d + d t o t 0 Z b l b2 +

fi(0

+

f ( 0

2 where = t e n s i l e s t r e s s due t o t h e t e n s i o n i n t h e Z w i r e rope, S,

d = i n t e n s i t y of s t r e s s due t o primary bending,

b 1

.L

a s d e f i n e d by Reuleaux o r I s a a c h s e n ,

fl = i n t e n s i t y o f s t r e s s due t o secondary bending

b2 r e s u l t i n g from t h e t r a n s v e r s e l o a d Po, f l ( 7 ) = s t r e s s due t o t w i s t , f 2 ( ? ) = s t r e s s due t o f r i c t i o n . The i n v e s t i g a t i o n d e s c r i b e d below d e a l s w i t h 6 b2

°

Taking i n t o account t h a t Po i s d i s t r i b u t e d o v e r a c e r t a i n d i s t a n c e , t h a t ' t h e span

1

i s s l i g h t l y reduced by t h e deforma- t i o n of t h e w i r e and t h a t t h e w i r e i s continuous, i t i s

assumed t h a t t h e maximum bending moment a t t h e c e n t r e of t h e f i e l d i s

M

max.

-

Po

J

Po

8

16 1 6 s i n @ where Po = t r a n s v e r s e l o a d of a w i r e ,

6

=' d i a m e t e r of a w i r e , O = t h e i n t e r s e c t i n g a n g l e o f two l a y e r s of w i r e s , T h i s bending moment was d e l i b e r a t e l y assumed t o be s m a l l e By i n t r o d u c i n g t h e moment o f r e s i s t a n c e of t h e outermost

w i r e

-

assuming t h a t t h e w i r e s of t h e two l a y e r s have i d e n t i c a l d i a m e t e r

-

t h e f o l l o w i n g i s o b t a i n e d :

The magnitude of t h e f o r c e Po, which depends on t h e manner i n which t h e t r a n s v e r s e l o a d i s a p p l i e d , i s a n i m p o r t a n t f a c t o r . The c o n s t r u c t i o n o f t h e w i r e r o p e ,

e x p r e s s e d by t h e a n g l e of l a y , w , and t h e t h i c k n e s s o f w i r e ,

8 ,

a r e a l s o i m p o r t a n t ,

1. Wire Rope

-

a s Defined by Reuleaux

-

C l o s e l y F i t t e d

a Over a Sheave

I n t h i s c a s e a n approximate formula f o r t h e f o r c e s Po may be o b t a i n e d from t h e t r e a d s shown i n F i g . 2, t a k i n g i n t o

a c c o u n t t h e width of t h e sheave grooves ( I o r 11) and t h e l a y of t h e r o p e , Po = f o r c e a c t i n g t r a n s v e r s e l y on a s i n g l e w i r e , S = t e n s i o n i n t h e w i r e r o p e , z = number of s t r a n d s i n t h e r o p e , m = number of w i r e s i n t h e r o p e

-

assuming t h a t t h e w i r e s a r e i d e n t i c a l , a = d i s t a n c e between t h e crowns ( A , B, C , e t c . ) of t h e s t r a n d s (a = L / z ) , n = number of p o i n t s o f c o n t a c t w i t h i n t h e d i s t a n c e a , L = l e n g t h of l a y of t h e s t r a n d s ,

D

= d i a m e t e r of sheave, d = d i a m e t e r of w i r e r o p e , t h e n

With r e s p e c t t o n i t i s assumed t h a t f o r t i g h t groove i n t h e sheave Po max, e q u a l s a mean v a l u e , which i s o b t a i n e d when t h e p r e s s u r e d i s t r i b u t i o n o v e r t h e w i d t h d/2 i s uniform,

I n t h i s c a s e

a p p l i e s f o r r e g u l a r l a y r o p e s and may a l s o be a p p l i e d w i t h good approximation t o l a n g l a y r o p e s , Hence, a s t h e mean v a l u e , whence, by e q u a t i o n ( 3 ) , For w i r e r o p e s c o n s i s t i n g o n l y of i d e n t i c a l w i r e s t h e t e n s i l e s t r e s s may be s u b s t i t u t e d , Then, f o r m w i r e s e q u a t i o n ( 5 ) becomes

= 2Lm

1

C Z

.

6b2 s i n o d z D

From t h i s i t i s evgdent t h a t t h e secondary bending, b b 2 , depends on t h e c o n s t r u c t i o n of t h e r o p e , t h e l e n g t h of l a y o f t h e s t r a n d s and w i r e s , t h e v a l u e _{6 / ~ and on} 6,. It d e c r e a s e s a s t h e l e n g t h of l a y d e c r e a s e s ,

For i n s t a n c e , i f L = 7,5d, z = 6 and w = BOO, t h e

m = 6 x 1 9 = 114 w i r e s ,

ob2

= 570 cZd/D ( 6 a ) Ill = 6 X 37 = 222 w i r e s , 6 b 2 = 1 1 1 0 f l Z d / ~ _{( 6 b )}

m

= 6 x 6 1 = 366 w i r e s , #b2 = 1830 @ z s / D ( 6 ~ ) For L = 8.0d t h e v a l u e s f o r Ob2 w i l l be g r e a t e r by a p p r o x i m a t e l y 6%. It w i l l be found t h a t t h e v a l u e s f o r r o p e s of mountain r a i l w a y s a r e s i m i l a r (assuming normal l a y , o f c o u r s e ) , I n a c t u a l c a s e s t h e i n t e r s e c t i n g a n g l e may be c o n s i d e r a b l y s m a l l e r , t h e r e b y g r e a t l y i n c r e a s i n g t h e a t r e s s e s due t o secondary bending, C b2 The f a c t t h a t t h e s t r e s s e s due t o 6 do p l a y a n b2

i m p o r t a n t r o l e i s e v i d e n t from F i g , 3, where N, t h e number of bends t o c a u s e f a i l u r e of t h e r o p e , has been p l o t t e d a s o r d i n a t e , These d a t a were o b t a i n e d from f a t i g u e t e s t s made by Vioernle and Herbst (1) w i t h r o p e s of v a r i o u s c o n s t r u c t i o n . Here t h e r o p e s have been a r r a n g e d a c c o r d i n g t o D/ 6

,

t h i s r a t i o v a r y i n g between 100 and 1070, Furthermore, t h e t e n s i l e s t r e s s ,

,

t h e t e n s i l e s t r e n g t h of t h e wire and t h e sum C t o t o l

z

= O J

+

cbl a r e g i v e n i n kgm, p e r s q , mm, F i g o 3 a l s o shows t h e l a y of t h e r o p e s ( l a n g l a y r o p e , r e g u l a r l a y r o p e o r S e a l e r o p e ) , t h e r o p e d i a m e t e r and t h e c o n s t r u c t i o n of t h e s t r a n d s , b u t t h e r e a r e no d a t a on t h e l e n g t h of l a y of t h e s t r a n d s and w i r e s , A s i s e v i d e n t from t h e f i g u r e , t h e shape of t h e N l i n e i s markedly u n d u l a t e , There i s o n l y a rough g e n e r a l

r e l a t i o n s h i p between D/$ and t h e number o f dends t o f a i l u r e , N, n o t t a k i n g i n t o a c c o u n t t h e c o n s t r u c t f on of t h e r o p e , I t c a n e a s i l y be s e e n t h a t t h e f a c t o r N i s v e r y low f o r v a l u e s of

D/S

between 100 and 300 and t h a t o n l y f o r h i g h v a l u e s o f ~ / 6 does i t g r a d u a l l y i n c r e a s e . The shape o f t h e c u r v e f o r i s a l s o noteworthy. F o r v a l u e s of ~ / 6 r a n g i n g from 100 t o 300 t h i s curve a s c e n d s above t h a t f o r

P,

( t e n s i l e s t r e n g t h ) a t some p o i n t s , I t i s not s u r p r i s i n g , t h e r e f o r e , t h a t under t h e s e c o n d i t i o n s t h e v a l u e s f o r N a r e v e r y low, The s t r e s s e s Fz 4- a l o n e do n o t s u f f i c e t o b l e x p l a i n t h e marked u n d u l a t e shape of t h e N-N c u r v e , However, i f t h e s t r e s s e s due t o secondary bending a r e t a k e n i n t o a c c o u n t , i , e o , i f t h e sum d t o t o 2 = 0, +

cbl

+

0' i s formed, i t w i l l be found t h a t t h e l o c a l

b2

d e s c e n t s of t h e N c u r v e always c o i n c i d e w i t h marked a s c e n t s of t h e c u r v e f o r

utoto2

.

This c l e a r l y d i s p l a y s t h e e f f e c t of Gb and i t becomes c l e a r now why t h e N-N

2

c u r v e must be u n d u l a t e f o r some t y p e s of r o p e and i n i t i a l s t r e s s e s , A number of s t r a n d c r o s s - s e c t i o n s a r e shown i n F i g , 3 f o r t h e w i r e r o p e s w i t h hemp c o r e employed i n t h e s e t e s t s ,

For S e a l e r o p e s , which d i f f e r from o t h e r r o p e

t y p e s by t h e f a c t t h a t t h e w i r e s w i t h i n a s t r a n d do n o t i n t e r s e c t ( c f , F i g , 3a, l a s t c r o s s - s e c t i o n ) , o n l y

t o t o l = Cz

+

_{b l} need be t a k e n i n t o a c c o u n t s i n c e 0' i s z e r o , High N v a l u e s a r e t h e r e f o r e o b t a i n e d f o r b2 t h e s e r o p e s , I n F i g o 3 t h e r o p e s i n q u e s t i o n a r e marked by a n S on t h e l i n e "wire l a y e r s t ' . S i n c e , a s e q u a t i o n ( 6 ) shows, i s d i r e c t l y pro- b2 p o r t i o n a l t o Cz, t h e l o n g i t u d i n a l s t r e s s has a g r e a t e f f e c t on t h e f a t i g u e f a c t o r N o T h i s i s noteworthy and i s a simple e x p l a n a t i o n of t h e f a c t t h a t N d e c r e a s e s a s 0 _z i n c r e a s e s ,

For l o o s e sheave grooves t h e s t r e s s e s due t o secondary bending become i n c r e a s i n g l y more u n f a v o u r a b l e , s i n c e i n

t h i s caae t h e number of p o i n t s o f c o n t a c t i s reduced con- s i d e r a b l y and Po i n c r e a s e d a c c o r d i n g l y . T h i s i s e v i d e n t from F i g , 4 which shows t h e r e s u l t s from a number of

t e s t s made by Woernle ( 2 ) w i t h r e g u l a r l a y r o p e s of 1 6 mm, d i a m e t e r and w i t h t i g h t and l o o s e sheave g r o o v e s , The e f f e c t which t h e i n c r e a s e d i n i t i a l l o a d of dz = 30 kgm, p e r s q , mm, has on t h e number of bends t o f a i l u r e i s e v i d e n t from t h i s c u r v e , The l a t t e r i s c o n s i d e r a b l y lower t h a n t h a t f o r

ez

= 10 kgm, p e r s q , mm, It i s a l s o e v i d e n t from F i g o 3 t h a t , a l t h o u g h db2 i s z e r o , t h e r e s t i l l i s c o n s i d e r a b l e s c a t t e r i n t h e N v a l u e s f o r t h e S e a l e r o p e s , This i s due p r i m a r i l y t o t h e compression a l o n g t h e s t r a n d s i n c o n t a c t , T h i s w i l l be d i s c u s s e d i n t h e subsequent s e c t i o n . Moreover, i t must be emphasized t h a t t h e above v a l u e s f o r

cb2

a r e

o n l y approxfmate v a l u e s , They m e r e l y s e r v e t o show t h e e f f e c t which secondary bending c a n have on t h e l i f e o f normal s t e e l w i r e r o p e s ,

2 , Rope

-

a s Defined by I s a a c h s e n

-

T r a n s v e r s e l y S t r e s s e d by a P u l l e x

Thfs t y p e of s t r e s s i s found i n cableways above t h e p u l l e y s of t h e p a s s f n g p l a c e s , A t t h e most u n f a v o u r a b l e p o f n t t h e t r a n s v e r s e f o r c e i s d i s t r i b u t e d h e r e o v e r two

o r p o s s i b l y t h r e e w i r e s , The p r e s s u r e p e r w i r e t h e n becomes Po-Q/2, The s e c o n d a r y bending i s o b t a i n e d from e q u a t i o n ( 3 ) a s where Q = f o r c e of t h e p u l l e y , = w i r e d i a m e t e r . i n t e r s e c t i n g a n g l e of t h e two o u t e r m o s t l a y e r s of w i r e s , For i n s t a n c e , f o r a t r a n s v e r s e f o r c e of 100 kgm,, a = 30' and s u r r o u n d i n g w i r e s of 3 , 5 rnm. d i a m e t e r t h e s e c o n d a r y bending, d b i s 5,25 kgm, p e r s q , mrn, I f 2 9 S e a l e r o p e s a r e used, (332 becomes z e r o h e r e t o o , B, S t r e s s due t o I n t e r n a l Compression

The problem of compressions i n s i d e t h e w i r e r o p e 9

h a s been r e s t r i c t e d h e r e t o two s p e c i a l c a s e s .

%-

1, F i e l d s of Force i n t h e C r o s s - s e c t i o n a l Area of a Wire An approximate i d e a of t h e f i e l d s of p r i n c i p a l s t r e s s e s i n l a t e r a l l y s t r e s s e d w i r e s may be o b t a i n e d from t h e example of d i s c s s t r e s s e d c o r r e s p o n d i n g l y o F i g u r e 5a shows t h e f i e l d of p r i n c i p a l s t r e s s e s of a c i r c u l a r d i s c t o which two f o r c e s , P, c o u n t e r a c t i n g each o t h e r a r e a p p l i e d , This diagram resembles somewhat t h e m i d s e c t i o n t h r o u g h t h e f i e l d of f o r c e of a wire where t h e l o a d i s a p p l i e d a t two o p p o s i t e p o i n t s and i s c h a r a c t e r i z e d by t h e f a c t t h a t a l l t h e l i n e s i s s u i n g from a p o i n t o f a p p l i c a t i o n o f t h e f o r c e P l e a d t o t h e o p p o s i t e p o i n t , F i g , 5b may be t a k e n approximately f o r a m i d s e c t i o n of a wire which i s s t r e s s e d by P1 and i s supported by

two a d j a c e n t w i r e s and i n F i g , 5c by t h r e e a d j a c e n t w i r e s , I n t h e c a s e of F i g , 5d t h e r e would be many p o i n t s of

s u p p o r t a t t h e lower s i d e of t h e w i r e , From t h i s i t i s e v i d e n t t h a t each s i n g l e load P has a d e f i n i t e l o c a l f i e l d which may be bounded by l i n e s and s i n g u l a r p o i n t s , The f i e l d s a , b and d were determined on sheaves by means of p h o t o e l a s t i c experiments, whereas t h e f i e l d 5c was

9

d e r i v e d by t h e a u t h o r from t h e l i m i t i n g c a s e s b and d o A l l t h e s e f i g u r e s must be c o n s i d e r e d approximations;

of t h e w i r e s i n a r o p e p r o b a b l y a r e a g r e a t d e a l more comp- l i c a t e d ,

2 , F i e l d o f Force due t o Tension i n a S t r e s s e d S t r a n d e d Rope ( a ) F i e l d of f o r c e due t o a x i a l t e n s i o n ,

I f a s t r a n d e d r o p e w i t h hemp c o r e i s l o a d e d a x i a l l y , t h e n a l l t h e s t r a n d s t e n d t o s t r a i g h t e n t h e m s e l v e s , t h u s p r o d u c i n g a motion i n a r a d i a l d i r e c t i o n toward t h e c e n t r e , Mutual

h i n d r a n c e of t h e s t r a n d s produces t h e r a d i a l f o r c e s V , and t h e hexagon c o n s i s t i n g of t h e l i n e s of f o r c e K i s o b t a i n e d ( c f , F i g s , 6 and 8 ) , S i n c e t h e r e i s o n l y t h e hemp c o r e i n t h e c e n t r e , t h e f o r c e s V and K must be i n e q u i l i b r i u m w i t h e a c h o t h e r , C o n d i t i o n s a r e s i m i l a r i n t h e i n d i v i d u a l s t r a n d s where polygons, determined by t h e number of w i r e s p e r l a y e r , a r e o b t a i n e d a s l i n e s of f o r c e , Moreover,from i n d i v i d u a l end p o i n t s of t h e f o r c e polygons l i n e s of f o r c e t e n d toward t h e c e n t r e , The p o i n t s T, a t which two s t r a n d s come i n t o con- t a c t and t h r o u g h which t h e l i n e of f o r c e K p a s s e s , a r e note- worthy ( F i g , 6 b ) , If t h e compressive f o r c e K i s t o o g r e a t , t h e n i t w i l l be found t h a t under c e r t a i n c o n d i t i o n s t h e r o p e shows s e v e r e i m p r i n t s a t t h e s e p o i n t s ( F i g , 71, The r o p e i n q u e s t i o n i s a s p a n rope of a c a b l e r a i l w a y , I t i s 33 mrn, i n d i a m e t e r and c o n s i s t s of e i g h t s t r a n d s o f S e a l e c o n s t r u c t i o n and a hemp c o r e , The o u t e r w i r e s have a d i a m e t e r of 1,8 mm, The r o p e shows nunierous w i r e b r e a k s

resulting from too strong a tensile stress and too small a sheave at the outside, The extraordinarily deep imprints at the points where two individual strands come into con- tact and the resulting -breaks are also noteworthy (points T in Figo 6b), From this it is evident that under certain conditions failures in Seale ropes may be due to this

cause. Close examination of the imprints shows that they are arranged as shown in Figo 10c, with the points B, B' standing out distinctly,

(b) Determination of the forces V,

K

and Po due to axial tensile load at the points of contact of two strands, These forces as well as the compressive forces can be approximately determined, graphically (Fig, 8) and

numerically, the example being a rope consisting of six strands as shown in Figo 8 a 0 However, the derivation below is general, The line ABCDCt represents the projection of the centre line (axis) of a strand (Fig, 8b), With the aid of this figure V may be approximately determined from the crown zone CDCq (Figs, 8 a and ad), if the centre line is assumed to be a helical curve having a given pitch,

If

S = tension in the wire rope,

S f = tension of a strand,

L

= length of lay of a strand

(cog.

7.5 to 8.0 d), z = number of strands in the wire rope,

d = d i a m e t e r o f t h e w i r e r o p e , d l ' d i a m e t e r o f a s t r a n d , a i n c l i n a t i o n o f a s t r a n d t o t h e r o p e a x i s ( a n g l e o f l a y of t h e s t r a n d ) , r = r a d i u s of c u r v a t u r e i n t h e p l a n e of d e f l e c t i o n o f t h e h e l i c a l c u r v e of a s t r a n d a t D, t h e n , a c c o r d i n g t o F i g , 8 c , S l = S 9 Z cosw and from F i g , 8 d i t f o l l o w s t h a t hence F u r t h e r m o r e , i t f o l l o w s from t h e h e l i c a l c u r v e t h a t : f l ( d

-

el&)

s i n

a

= c o s (3 =

L

V L ~

+

n z

( d

-

d x ) 2 A c c o r d i n g l y , w i t h r e s p e c t t o u n i t of l e n g t h ,

v

= 2 s i n C% t a n a z(d

-

d l ) S o r , i f s i n a and tan& a r e r e p l a c e d by t h e d i m e n s i o n s o f t h e r o p e ,

The compressive f o r c e a t B o r B t , t h e p o i n t of c o n t a c t of two s t r a n d s (Figo 1 0 c ) , i s

where

and

For t h e p o i n t s C and A t t h e compressive f o r c e would o n l y be one h a l f t h a t a t B ( c f . F i g . 7 ) .

There a r e r e l a t i o n s h i p s between wire and s t r a n d

analogous t o t h o s e between s t r a n d and rope; t h e r e f o r e , t h e f o l l o w i n g symbols ( c f , F i g o 8 ) a p p l y t o a s t r a n d :

L t = l e n g t h of l a y of t h e s u r r o u n d i n g w i r e s ,

a t = d i s t a n c e between t h e crowns of t h e i n d i v i d u a l w i r e s ( F i g . l O c ) ,

dt = a n g l e between t h e a x i s of a wire and t h e a x i s of a s t r a n d ( F i g . l O c ) , z l = number of s u r r o u n d i n g w i r e s , r t = r a d i u s of c u r v a t u r e of a wire i n ' t h e p l a n e of d e f l e c t i o n , = s t r a n d diameter,

6

= w i r e diameter. Then, a c c o r d i n g t o F i g , 10c,

Furthermore, s i n as = n(d&

-

-

6 )

Ji'z

+ n z ( d l

-

6

I z

where t h e a d d i t i o n a l c u r v a t u r e due t o t h e b e n t s t r a n d i s n e g l e c t e d , From e q u a t i o n s ( 9 ) , ( 9 a ) and ( 9 b ) t h e f o l l o w i n g i s o b t a i n e d : which, by e q u a t i o n ( 8 c ) , becomes

-

Po

-

8

s i n a t a n a

s

( 9 d ) 2 s i n

a'

z cos ~ (

-

dd x )

For s i n wv

,

s i n

a,

t a n & and cos 'b t h e dimensions of t h e rope may be s u b s t i t u t e d ,

I f a i r ~ U , e q u a t i o n ( 9 d ) may be reduced t o

For a wire rope w i t h s i x s t r a n d s and l e n g t h of l a y of t h e s t r a n d , L

=

7,5d, t h e f r a l u e s ' o b t a i n e d a r e :

z

= 6,

Q = 1 5 ° 4 ~ 1 ,

g =

60°, d

-

d b = 2

3

d Furthermore, a9 =

a,

c o r r e s p o n d i n g c o n s t r u c t i o n of t h e s t r a n d assumed of c o u r s e ,

Then, by e q u a t i o n s ( 8 c ) and ( g a l ,

i s o b t a i n e d and,by e q u a t i o n ( 9 e ) , t h e c o m p r e s s i v e f o r c e p e r w i r e a t t h e p o i n t of c o n t a c t of two s t r a n d s due t o a x i a l t e n s i l e l o a d :

3 , F i e l d of F o r c e i n a T r a n s v e r s e l y Loaded S t r a n d e d Wire Rope and t h e Compressive F o r c e s a t t h e P o i n t s of

C o n t a c t o f Two S t r a n d s

-

( a ) E f f e c t o f a c o n c e n t r a t e d f o r c e P a c c o r d i n g t o I s a a c h s e n , I f i n t h e most u n f a v o r a b l e c a s e o f F i g , 9 a t r a n s v e r s e l o a d P i s a p p l i e d , t h e n i t may be assumed t h a t t h e r e i s a l o a d P/Z a c t i n g on t h e c e n t r e of e a c h s t r a n d of a w i r e r o p e h a v i n g z s t r a n d s , The r e s u l t i s a z p o l y g o n o f

K

f o r c e s ( F i g , 9 b ) s u c h a s i s o b t a i n e d i n a s i m p l i f i e d way f o r a s t r e s s e d w i r e r o p e which i s n o t l o a d e d t r a n s v e r s e l y , Regarding t h e d i s t r i b u t i o n of t h e l a t e r a l c o m p r e s s i o n s a l o n g a s t r a n d due t o t h e f o r c e P i t must be assumed t h a t t h e y w i l l be g r e a t e s t i n t h e r e g i o n of t h e p o i n t o f a p p l i c a - t i o n and t h a t t h e y w i l l f a d e t o w a r d s b o t h s i d e s i n a

l o n g i t u d i n a l d i r e c t i o n , I n o r d e r t o determ.ine t h e "peak v a l u e s " i t i s assumed t h a t t h e c o m p r e s s i o n due t o P a l o n g t h e l i n e o f c o n t a c t o f t h e s t r a n d i n q u e s t i o n i s u n i f o r m l y

d f s t r f b u t e d o v e r t h e l e n g t h

,

which i s assumed t o be e q u a l t o t h e s t r a n d d i a m e t e r d ~ , The h e a v i e s t i m p r i n t s a t t h e p o i n t T ( c f , F i g , 6 b ) o c c u r i n - t h e s t r a n d which i s d i r e c t l y l o a d e d by t h e f o r c e P o I n a w i r e r o p e w i t h z s t r a n d s t h i s f o r c e must t r a n s m i t a f o r c e

PI

= P ( z

-

l ) / z l a t e r a l l y , I n o r d e r t o combine t h i s f o r c e w i t h V due t o t h e s t r e n g t h of t h e w i r e r o p e , S,

P s

must be c o n v e r t e d i n t o u n i t of l e n g t h , Then t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d ;

and ,by e q u a t i o n s ( 9 ) , ( 9 a ) and ( 9 b ) , t h e c o m p r e s s i v e f o r c e a t t h e p o i n t of c o n t a c t of two s t r a n d s becomes o r , i f t h e s t r e n g t h of t h e w i r e r o p e , S, o f e q u a t i o n ( 8 c ) i s s u b s t i t u t e d f o r V,

-

Po

-

, 8 ( z

-

1 ) P + 4 s i n u - q c o s ' b

z

-!L F o r a w i r e r o p e h a v f n g s i x s t r a n d s and l e n g t h of l a y L = 7,5d, t h e f o l l o w i n g v a l u e s a r e o b t a f n e d : z = 6,

.

..

a =

1 5 ° 4 0 ~ ,

b

= 60° and U P =

a ,

assuming c o r r e s p o n d i n g c o n s t r u c t i o n o f t h e s t r a n d . F u r t h e r m o r e , ( d

-

dk) = 2d/3 and = d ~ '= d / 3 , . Thus,

If i t were assumed t h a t Po

,,,

r e s u l t from u n i f o r m p r e s s u r e d i s t r i b u t i o n o v e r a l e n g t h = d , t h e n where Po = compressive f o r c e p e r w i r e o f two s t r a n d s coming i n t o c o n t a c t , P = p r e s s u r e o f t h e s h e a v e , i , e , , a s i n g l e f o r c e a s d e f i n e d by I s a a c h s e n , S = t e n s i o n i n t h e w i r e r o p e ,

8

= w i r e d i a m e t e r o f t h e s u r r o u n d i n g w i r e s , d = d i a m e t e r o f w i r e r o p e , I f , c o n t r a r y t o F i g o 9b, t h e p r e s s u r e o f t h e s h e a v e i s d i s t r i b u t e d o v e r two o r t h r e e s t r a n d s , Po w i l l d e c r e a s e a c c o r d i n g l y , ( b ) Wire r o p e r u n n i n g o v e r a s h e a v e a s d e f i n e d by Reuleaux, According: t o e q u a t i o n ( 4 ) t h e f o r c e a c t i n g p e r cr"own of a s t r a n d i s P = 2aS/D, (4bb

The l o c a l d i s t r i b u t i o n o f compression due t o t h e f o r c e P a l o n g two s t r a n d s coming i n t o c o n t a c t c a n v a r y , d e p e n d i n g on t h e form of t h e s h e a v e groove, For c l o s e l y f i t t i n g

groove t h i s c o m p r e s s i o n s h o u l d b e u n i f o r m l y d i s t r i b u t e d o v e r t h e e n t i r e l e n g t h , i , e , , t h e l e n g t h = a = L / Z , For wide open groove, however, t h e g r e a t e s t l a t e r a l p r e s s u r e should r e s u l t a s t h e mean v a l u e from a u n i f o r m p r e s s u r e d i s t r i b u t i o n

over A = d/3, ( i ) For c l o s e l y f i t t i n g groove t h e f o l l o w i n g i s o b t a i n e d from e q u a t i o n s ( l l ) , ( 4 b ) and ( 8 c ) : K12 = 2 _{cos 5} ( z

-

1) 2 S + 2

_D

_{z ( d} s i n o t a n d

_-

_dX"} and from e q u a t i o n ( l l a ) ;

-

Po

-

_{22 s i n osqcosi} S

8

(

z

;

1 + s i n a t a n a ( d

-

dgl

Then, f o r a w i r e rope having s i x s t r a n d s and l e n g t h of l a y L = 7,5d, i f do = (3 _{and z = 6,} ( i i ) For v e r y l o o s e groove, t h e f o l l o w i n g i s o b t a i n e d : and, by e q u a t i o n . ( l l a ) ,

-

-

s

6

3 ( z

-

l ) a + s i n a t a n o c '0 z 2 s i n

a!

c o s y

I n t h i s c a s e , f o r t h e wire rope having s i x s t r a n d s and l e n g t h of l a y L = 7,5d, i f =

a,

where Po = compressive f o r c e p e r w i r e of two s t r a n d s coming i n t o c o n t a c t

,

S = t e n s i o n i n t h e wire rope, d = w i r e r o p e d i a m e t e r ,

.

D = sheave d i a m e t e r ,

Comparison of equations (15a) and (16a) clearly shows that Po increases considerably for very loose sheave groove, 4, Maximum Compression po on a Wire in the Elastic Range

Hertzvs formula for the maximum compressive stress, po, at the point of contact is (3):

4 / T , l ~ ,i'

where Po = local compressive force on a wire,

'P

= ~1 + ~2 + 8 3 + ~ 4 the sum , of the principal

curvatures,

/ U V = an auxiliary value which depends on the auxiliary angle 1

.

For steel and cast steel

E

= 20,000 kgm, per sq, mm, In the present case the following is obtained frorn Fig. 10:

cos

r =

1 ' 2

-~(?ll-?l2) + 2(?11- ?121(?21- p22 1 ~ 0 + ~

(y21-

2 ~

P

~

~

)

~

where a ) a t t h e p o i n t of c o n t a c t o f w i r e r o p e and s h e a v e : R~~ = r a d i u s of c u r v a t u r e of t h e c e n t r e l i n e o f a w i r e (approximately a l s o t h e r a d i u s o f c u r v a t u r e of t h e crown l i n e of a w i r e ) , H12 = r a d i u s of c u r v a t u r e o f a w i r e i n t h e c r o s s - s e c t i o n =

6'2,

RZ1 = r a d i u s o f t r e a d ( r u n n i n g r a d i u s ) o f a s h e a v e o r p u l l e y , R22 = r a d i u s of g r o o v e ,

"

= a n g l e b e t w e e n a w i r e and a t r e a d a t t h e b o t t o m of a s h e a v e , ( 1 n F i g , 1 5 t h i s a n g l e i s d e s i g n a t e d b y ~p 0 ) b! a t t h e p o i n t o f c o n t a c t of two s t r a n d s ( F i g , l o b ) : R1l = r a d i u s o f c u r v a t u r e o f t h e c e n t r e l i n e of a w i r e o f s t r a n d 1 ( a p p r o x i m a t e l y a l s o t h a t of t h e crown l i n e ) , C\ R12 = '/2 o f t h e c r o s s - s e c t i o n o f a w i r e i n s t r a n d 1, R21 = r a d i u s of c u r v a t u r e of t h e c e n t r e l i n e of a w i r e i n s t r a n d 2, R22 = &/2 o f t h e c r o s s - s e c t i o n o f a w i r e i n s t r a n d 2 ,

--

t h e a n g l e b e t w e e n two w i r e s i n c o n t a c t , The v a l u e pJ f s o b t a i n e d f r o m c o s r w i t h t h e a i d o f t h e c u r v e shown i n F i g , 11, Whence i t f o l l o w s t h a t c ) a t t h e p o i n t o f c o n t a c t of r o p e and s h e a v e (made o f c a s t s t e e l )

( i ) F o r r e g u l a r l a y r o p e s : R 1 1 2 r B = d l Q =

<

-*=

ti

R12 6 6 s i n 2 cxf p o s i t i v e where r P i s determined by e q u a t i o n ( 9 e ) , Furthermore, =

R22

n e g a t i v e R12 =

-

R21 =

D

p o s i t i v e R12

6

cJ

-

0, C O S 2w-1,o Taking i n t o a c c o u n t t h e s i g n s , t h e n o r , s i n c e R12 = 8/2 of t h e o u t e r w i r e s , F o r i n s t a n c e , assume t h a t f o r r u n n i n g p u l l e y s p v - l , 5 , M -0.94 and f o r ( c a r r y i n g ) s h e a v e s ~ ~ ~ l . 6 , Id -0.92. I n o r d e r t o o b t a i n a comparative v a l u e , t h e f o l l o w i w

v a l u e s a r e s u b s t i t u t e d i n e q u a t i o n (20a):/ccv = 1 , 6 and M = O09;5, Then t h e f o l l o w i n g mean v a l u e i s o b t a i n e d :

-

c=l%67F

Po

-

( 2 W "

3

According t o Reuleaux, e q u a t i o n ( 4 a ) a p p l i e s t o po f o r t i g h t grooves i n t h e sheave.

( i i ) F o r l a n g l a y r o p e s o, rn and n have t h e same s i g n s a s f o r r e g u l a r l a y r o p e s , b u t o-27' and 2 c o s 2 ~ - 1 , 2 . T h e r e f o r e , 1 1

r

=

{

l + - + - - ~ \ o n m F o r c a s t s t e e l p u l l e y s , i f R12 = 812, Assuming t h e f o l l o w i n g v a l u e s a s an example f o r r u ~ n i n g p u l l e y s : p V - - 1 , 7 , N!-0.94 / and f o r ( c a r r y i n g ) s h e a v e s : p v - 2 . 2 , hlcO .92, t h e n w i t h ~ e a n v a l u e s o f p v - 2 , O and h!+0,93, From t h i s i t i s e v i d e n t t h a t a t t h e s u r f a c e c o m p r e s s i o n s i n 1an.g l a y r o p e s a r e s l i g h t e r t h a n i n r e k u l a r l a y r o p e s , Po i s o b t a i n e d from e q u a t i o n ( 4 a ) , d ) A t t h e p o i n t o f c o n t a c t o f two s t r a n d s

A s Fig, l o b shows, o, rn a n d n become p o s i t i v e h e r e and o --/ 30' f o r b o t h r e g u l a r l a y r o p e and l a n ~ l a y r o p e , From symmetry c o n s i d e r g t i o n s i t f o l l o w s t h a t ( 2 2 1

-

i

;-+

,/

ECOSU

= I - -

C O S T

= O c o s w * 1 1 ( 2 3 ) 1 9 - 0 1

-+

-

₀ F o r s t r a n d e d w i r e r o p e s l / o = l i e s b e t w e e n

t h e a n k l e between t h e i n t e r s e c t i n L w i r e s of' s t l + a l ~ d 1 and 2 ,

/ U v l i e s between 1.20 and 1 , 4 0 , I t s mean i s 1.J. By sub- s t i t u t i n g Lhe v a l u e s b! = 2.04 a n d p g =

1,s

i n e q u a t i o n ( 2 0 a ) , t h e @ o n p r e s s i o c between two s t r a x d s coming i n t o c o n t a c t

A c c o r d i n g t o Reuleaux e q u a t i o n ( 1 8 ) a p p l i e s f o r Po. F i g . 1 2 shows t h e r e l a t i o n s h i p s between Po,

8

and p,. The l a t t e r c a n be r e a d d i r e c t l y w i t h t h e a i d of Po,

Comparison o f t h e c ~ m p ~ e s s i v e s . t r ~ e s s e s , p o , f o r p o i n t s of c o n t a c t o f two s t r a n d s and t h o s e f o r p o i n t s of c o n t a c t of r o p e and p u l l e y ( o r r o p e a n a s h e a v e ) shows t h a t t h e @om- p r e s s i v e s t r e s s e s between two s t r a n d s a r e c o n s i d e r a b l y g r e a t e r

( c f , e q u a t i o n s ( 2 0 b ) , ( 2 1 a ) and ( 2 4 ) a s w e l l a s T a b l e s V I and

VII)

0

5, The P r i n c i p a l S t r e s s e s and t h e Comparison S t r e s s e s ac t h e C e n t r e of t h e I m p r i n t e d S u r f a c e ( a ) P r i n c i p a l s t r e s s e s I n t h e e l a s t i c r e g i o n t h e p r i n c i p a l s t r e s s e s a r e (, :- ) o b t a i n e d f r o m t h e f o l l o w i n g s t r e s s e s : ( l o n g i t u d i n a l s t r e s s ) ( t r a n s v e r s e s t r e s s ) ( v e r t i c a l s t r e s s ) _{Q3 =}

-

_3Po

_-

-

P o ' ( 2 5 ~ ) 2n a b (-2 ) c f , r e f , 3 , p , 281.

where Po = compressive s t r e n g t h o f a n e x t e r n a l f o r c e i n kgm,, e = r a t i o of t h e e l l i p t i c a l a x e s b/a of t h e i m p r i n t e d s u r f a c e , m = P o i s s o n ' s r a t i o @ 1 0 / 3 po = compression a t t h e c e n t r e o f t h e i m p r i n t e d s u r f a c e i n kgm. p e r s q , rnm, A s e q u a t i o n ( 2 5 ) shows t h e p r i n c i p a l s t r e s s e s

rl

and

@2 g r e a t l y depend on t h e r a t i o b/a. T h i s i s a l s o e v i d e n t from T a b l e I ,

The t h r e e p r i n c i p a l s t r e s s e s ( c19 6 2 and 0 3 ) f o r which

t h e i m p r i n t e d s u r f a c e i s a c i r c l e have b e e n l i s t e d f o r t h e r a t i o a/b =

l o o u

I f a/b = 009 t h e n t h e r e i s c o n t a c t a l o n g a c u r v e , T h i s i s t h e o t h e r l i m i t i n g c a s e , (b) The comparison s t r e s s From t h e t h r e e p r i n c i p a l s t r e s s e s C1, r2 and

c3

t h e comparison s t r e s s i s o b t a i n e d a s

The s t r e s s

O;F

o c c u r s i n t h e t e n s i o n zone o f t h e w i r e w i t h c o r r e s p o n d i n g s e c o n d a r y bending and i n t h e p r e s s u r a e zone w i t h c o r r e s p o n d i n g l o c a l c o m p r e s s i o n , ( i ) T e n s i o n zone Because o f t h e u n i a x i a l s t a t e of s t r e s s a t t h e crown o f t h e c r o s s - s e c t i o n a l a r e a o f t h e w i r e , . t h e comparison s t r e s s becomes ( i i ) P r e s s u r e zone

I n t h i s zone Cl,

e2

and

cs

due t o Po ( e q u a t i o n ( 2 5 ) ) a c t on t h e s u r f a c e , and Wz and

-

Wb a c t l o n g i t u d i n a l l y , The l o n g i t u d i n a l p r i n c i p a l s t r e s s t h e r e f o r e becomes I f t h e s e a d d i t i o n a l s t r e s s e s a r e t a k e n i n t o a c c o u n t , t h e n from t h e c o e f f i c i e n t s l i s t e d i n T a b l e I t h e f o l l o w i n g g e n e r a l e q u a t i o n f o r t h e comparison s t r e s s a t t h e p o i n t o f i m p r i n t o f t h e w i r e s u r f a c e i s o b t a i n e d : I Table I1 g i v e s t h e n u m e r i c a l v a l u e s f o r x2 and y f o r v a r i o u s v a l u e s of b / a o E q u a t i o n ( 2 8 ) may a l s o be w r i t t e n i n a p p r o x i m a t e l y t h e f o l l o w i n g f orm:

Then, w i t h po, 6, and

rb

a s a b s o l u t e v a l u e s , t h e f o l l o w i n g h e n e r a 1 e q u a t i o n i s o b t a i n e d : The xs v a l u e s a p p l y i n g h e r e a r e l i s t e d i n Table 11, A s c a n be s e e n t h e d e v i a t i o n i s g r e a t e s t l'or b/a =

l o o o

Table 11, A t t h e i n s i d e of a w i r e below t h e c e n t r e of t h e i m p r i n t e d a r e a t h e r e i s a p o i n t where t h e s t r e s s C i s c o n s i d e r a b l y g (-3) g r e a t e r ( 4 ) . From compressive s t r e s s a l o n e t h i s s t r e s s becomes f o r b/a = 1, 0 = Oo62 po 63 and f o r b/a = 0, = 0.55 p,

.

g 6 , T r a n s v e r s e Loading T e s t s i n t h e P l a s t i c Region w i t h S t e e l Wires

The e x p e r i m e n t s d e s c r i b e d 'below were c a r r i e d o u t by t h e *

( -:: ) _{c f , r e f ,}

EMPA and were i n t e n d e d t o d e t e r m i n e ( i ) t h e r a t i o a/b of t h e p r i n c i p a l a x e s of t h e i m p r i n t e d a r e a f o r v a r i o u s c u r v a t u r e c o n d i t i o n s . and ( i i ) t h e maximum c o m p r e s s i v e f o r c e s i n t h e p l a s t i c r e g i o n , The e x p e r i m e n t a l d a t a a r e l i s t e d below. 1, Wires o f 3,42 mrn, diam. w i t h Cs = 1 2 3 kgrn. p e r s q . m a ,

Az

= 1 5 8 kgrn. p e r s q . mm.

2 a Vi'Jires of 4.21 rnrn, diam. w i t h

cs

= 1 2 0 kgm. p e r s q . mmes

p,

= 1 5 4 kgrn. p e r s q . mm.

Y o Hardened s t e e l p u l l e y s o r s h e a v e s w i t h r a d i i R21 = 29, 1 5 0 , 250, 500 and 1 , 0 0 0 rnm,, and g r o o v e r a d i i R22 = 1 5 , 4 0 mm,

and i n f i n i t e , I n two c a s e s t h e p u l l e y ( o r s h e a v e ) was f i x e d a t a n g l e s of l e s s t h a n 18' t o t h e w i r e a x i s , i . e . , a t a p o s i t i o n s i m i l a r t o t h a t i n l a n g l a y r o p e o r s p i r a l wound w i r e r o p e . F u r t h e r m o r e , t h e f o l l o w i n g v a l u e s were u s e d : 4 , Compressive f o r c e s

Po

from 1 0 0 t o 4,000 kgm. 5. R a d i i of c u r v a t u r e o f t h e w i r e s , Rll = 100, 200 mm. and i n f i n i t e . I t s h o u l d be remembered h e r e t h a t a c c o r d i n g t o e q u a t i o n s ( 8 a ) and ( 8 b ) t h e r a d i u s Rll r e l a t i v e t o t h e s u r r o u n d i n g w i r e s i s R1l = L~ + n 2 ( d f o r t h e s p i r a l wound r o p e 2 n 2 ( d

-

6 )

and R 1 1 7 L ' ~ + x 2 ( d l

-

6

I 2

f o r t h e s t r a n d e d r o p e ( E q . 3 c )

The e x p e r i m e n t a l r e s u l t s a r e shown i n F i g s . 13, 1 4 and 1 5 . F i g . 1 3 shows t h e i m p r i n t e d a r e a of a w i r e o f 4.2 mrn. d i a m e t e r . T h i s i m p r i n t was c a u s e d b y a p u l l e y o f 58 mm. d i a m e t e r i m p r e s s e d w i t h a f o r c e of 4,000 kgm, The w i r e had a c o u n t e r c u r v a t u r e of 200 rrlm, r a d i u s , and t h e g r o o v e r a d i u s a t t h e c i r c u m f e r e n c e of t h e p u l l e y was 1 5 mm. The r a t i o a/b i s a p p r o x i m a t e l y 3 , F o r l o w e r c o m p r e s s i v e f o r c e s t h e r a t i o s were found t o be a n a l o g o u s .

The i m p r i n t shown i n F i g . 1 4 was c a u s e d by a s h e a v e o f 2,000 mm. d i a m e t e r w i t h a c o m p r e s s i v e f o r c e of 3 , 0 0 0 kgm.

A w i r e of t h e same c u r v a t u r e and a d i a m e t e r of 4.2 mm. was u s e d , The g r o o v e r a d i u s was 1 5 mm, The r a t i o a/b i s a p p r o x i m a t e l y 1 0 i n t h i s c a s e and d i f f e r s b u t l i t t l e f o r l o w e r l o a d s ,

I n o r d e r t o d e t e r m i n e t h e mean c o m p r e s s i v e f o r c e s a l l t h e i m p r i n t e d a r e a s w e r e measured w i t h 20 f o l d m a g n i f i c a t i o n w i t h t h e a i d o f a.n i n t e g r a t o r . The r e s u l t i s n o t e w o r t h y .

I t was f o u n d t h a t i n t h e p l a s t i c r e g i o n t h e v a l u e pomean remained a l m o s t c o n s t a n t t h r o u g h o u t and i n d e p e n d e n t of t h e c o m p r e s s i v e f o r c e Po and t h e c u r v a t u r e c o n d i t i o n s . The e x t e n t of t h e

i m p r i n t e d a r e a depended on t h e v a l u e po d i r e c t l y ( F i g . 1 5 ) . The r e s u l t s may be summarized a s f o l l o w s :

1, R a t i o s of a x e s of t h e i m p r i n t e d a r e a s .

The r a t i o s d e t e r m i n e d a r e l i s t e d i n T a b l e 111. The c i t e d r a t i o s of a x e s o c c u r even f o r v e r y s l i g h t c o m p r e s s i v e f o r c e s

Po and s h o u l d a l s o be a p p l i c a b l e i n t h e e l a s t i c r a n g e , T a b l e 111, R a t i o s of a x e s of t h e i m p r i n t e d a r e a s .

I

2 , Mean and maximum c o m p r e s s i v e f o r c e .

A c c o r d i n g t o F i g , 1 5 t h e mean c o m p r e s s i v e f o r c e i s a l m o s t c o n s t a n t , and w i t h r e s p e c t t o t h e w i r e i t must be e x p e c t e d t h a t

A,

= 1 5 4 kgm. p e r s q . mrn. Hence R a d i u s of p u l l e y o r s h e a v e , R 2 1 9 i n mm, 30 > I 5 0 t o 1 , 0 0 0 > I 5 0 t o 1 , 0 0 0 > I 5 0 t o 1 , 0 0 0

-

227 kgm, p e r s q , mm, Po mean

-

( 3 1 o r

-

l 0 4 5 P z

.

T h i s must be c o n s i d e r e d a low v a l u e , An e x p l a n a t i o n w i l l be g i v e n u n d e r 3 , R a d i u s o f k r o o v e , R22

'

i n mm, 1 5 40 and 00 40 and oo 40 and a, R a d i u s of c u r v a t u r e of t h e b e n t wire9 Rll 9 i n mm, 1 0 0 t o 00 1 0 0 200 w I Assuming t h a t d e s p i t e t h e p l a s t i c r e g i o n i n c r e a s e d R a t i o of a x e s a/b a p p r o 3 a p p r . 1 0 a p p r , 1 2 1 2 t o 29 t e n s i o n i s p o s s i b l e i n t h e c e n t r e o f t h e i m p r i n t e d a r e a , t h e f o l l o w i n g i s s u b s t i t u t e d a s t h e maximum c o m p r e s s i v e f o r c e a c c o r d i n g t o e q u a t i o n ( 2 5 ~ ) : Po max, = l 0 5 p o mean = 340 kgm, p e r s q . mm, o r 2 . 1 5 ~ z o

T h e r e f o r e , po

,,,,

-

2/, may be c o n s i d e r e d t h e c r i t i c a l i n t e n s i t y of c o m p r e s s i v e s t r e s s on t h e s u r f a c e , On r e a c h i n g t h i s c r i t i c a l v a l u e t h e s u r f a c e of t h e w i r e w i l l show p e r - manent i m p r i n t s s u c h t h a t t h e mean i n t e n s i t y o f c o m p r e s s i v e s t r e s s g e n e r a l l y 'becomes Po mean 4 0 4 5 / 3 z o 3, R e l a t i o n s h i p b e t w e e n t h e c o m p a r i s o n t e n s i o n and Q

t h e v a l u e s po mean and Po ma,, i n t h e r e g i o n o f t h e e l a s t i c - p l a s t i c l i m i t , I f t h e m a t e r i a l i s p l a s t i c a l l y deformed, t h e n

@;-"s

9 ( 3 2 ) p r o v i d e d t h a t o' i s t a k e n a s t h e l i m i t of c o m p r e s s i o n due 9 t o u n i a x i a l s t r e s s i n t h e homogeneous m a t e r i a l , F u r t h e r m o r e , i t may b e assumed t h a t i n s t e e l w i r e s 0' -0,75p 4 Z (33) These a r e t h e c o n d i t i o n s u n d e r which t h e r e l a t i o n s h i p s g i v e n i n T a b l e I V e x i s t f o r t h e v a r i o u s i m p r i n t e d a r e a s and t h e c o m p r e s s i v e f o r c e po i n t h e p l a s t i c r e g i o n f o r t h e p r e s s u r e zone a t t h e s u r f a c e and a t t h e i n s i d e ,

T a b l e I V , Comparison t e n s i o n s i n t h e -. p r e s s u r e r e g i o n . -.- I n t h e f i r s t two columns t h e v a r i o u s i m p r i n t e d a r e a s w i t h t h e c o r r e s p o n d i n g r a t i o o f t h e p r i n c i p a l a x e s a r e l i s t e d , t h e c i r c l e and s t r i p b e i n g l i m i t i n g c a s e s , The c o r r e s p o n d i n k , v a l u e s of d f o r t h e p r e s s u r e zone a t t h e s u r f a c e and a t t h e i n s i d e , a s 43 o b t a i n e d f r o m e q u a t i o n (28d) i f bZ - C _b = 0, a r e shown i n t h e t h i r d and t h e f i f t h c o l u m n s , The r e s u l t i n g v a l u e s o f po i n t h e p l a s t i c r e ~ i o n a r e o b t a i n e d f r o m e q u a t i o n s ( 3 2 ) and ( 3 3 ) : where xs i s t h e f a c t o r f o r t h e v a r i o u s f o r m s o f i t r n p r i n t l i s t e d i n T a b l e 11, Then, f o r a n e l l f p s e w i t h a / b = 3 i n t h e p l a s t i c

r e g i o n po = 2.77PZ a t t h e s u r f a c e , w h e r e a s st t h e i n s i d e p o = 1 . 2 5 P z O For t h e o t h e r c a s e , i . e . , where a/b = 1 0 , po = 2.14/8, a t t h e s u r f a c e and p o = 1 . 2 9 P Z a t t h e i n s i d e . From t h e e x p e r i m e n t s d i s c u s s e d i n t h e p r e s e n t s e c t i o n a v a l u e of p o mean = l 0 4 5 f l Z was o b t a i n e d t h r o u b h o u t , i r r e s p e c t i v e o f a / b , From t h i s i t i s o b v i o u s t h a t w i t h t h e s e c o m p r e s s i v e f o r c e s i n t h e p l a s t i c r e g i o n t h e s t r e s s c o n d i t i o n s a t t h e i n s i d e become o f p r i m e i m p o r t a n c e , T h i s shows ( i ) i n t h e low e x p e r i m e n t a l v a l u e s f o r p o mean, w h i c h a r e o n l y s l i g h t l y h i g h e r t h a n t h e t h e o r e t i c a l v a l u e s i n s i d e t h e p r e s s u r e zone, and ( i i ) i n t h e c o n s t a n c y of t h e e x p e r i - m e n t a l v a l u e s f o r po mean, s i n c e t h e x" v a l u e s a t t h e i n s i d e a l s o a r e a l m o s t i n d e p e n d e n t o f t h e f o r m of t h e i m p r i n t e d a r e a a s d e f i n e d by e q u a t i o n s ( 2 9 ) and ( 3 0 1 , F o r a c r i t i c a l e s t i m a t e of t h e p l a s t i c r e ~ i o n i t may be assumed, t h e r e f o r e , t h a t po mean = 1 , 4 5 / j z 0 S i n c e t h e w i r e s i n v o l v e d a r e p r e - s t r e s s e d , i t i s n o t i m p o s s i b l e t h a t t h e B a u s c h i n g e r e f f e c t w i l l a l s o make i t s e l f f e l t a t t h e p o i n t s o f i m p r i n t and w i l l c o n t r i b u t e t o t h e r e l a t i v e l y e a r l y a p p e a r a n c e o f p e r m a n e n t deforma- t i o n s ,

hioreover, i t must be emphasized t h a t a l l t h e r e s u l t s , r e f e r r i n g t o t h e p l a s t i c r e g i o n , a r e m e r e l y d a t a of a n i n f o r m a t i v e c h a r a c t e r and t h a t t h o r o u g h i n v e s t i g a t i o n s , t h e o r e t i c a l a s w e l l a s t e c h n i c a l , a r e r e q u i r e d t o c l a r i f y

c o n d i t i o n s a c c u r a t e l y .

7. F a t i g u e T e s t s w i t h Wires o f V a r i o u s S u r f a c e C o n d i t i o n s , Bending and t o r s i o n f a t i g u e t e s t s w i t h s i n g l e w i r e s of 3.4 and 4,2 mm. d i a m e t e r and v a r i o u s s u r f a c e c o n d i t i o n s were c a r r i e d o u t by t h e EMPA. The f o l l o w i n & w i r e s were t e s t e d a t one m i l l i o n c y c l e s o f t h e l o a d : a ) w i r e s w i t h normal s u r f a c e c o n d i t i o n s , b ) w i r e s w i t h l o c a l i m p r i n t s r e s u l t i n g from l o a d s of 250, 500 and 1,000 kgm,, c ) w i r e s w i t h permanent t w i s t , d ) w i r e s showing s i g n s o f c o r r o s i o n , e t c ,

The r e s u l t s a r e shown i n F i g . 1 6 where t h e f a t i g u e s t r e n g t h i n kgrn. p e r s q , rnm, a t one m i l l i o n c y c l e s has been p l o t t e d v e r t i c a l l y , The w i r e s l i s t e d above under a ) t o d ) a r e g i v e n i n o r d e r from l e f t t o r i g h t , From t h i s f i g u r e i t i s e v i d e n t t h a t when t h e q u a l i t y o f t h e w i r e m a t e r i a l i s s a t i s - f a c t o r y and t h e l o c a l i m p r e s s i o n s due t o h a r d e n i n g o f t h e m a t e r i a l a r e n o t t o o s e v e r e , t h e f a t i g u e s t r e n g t h shows no marked d e c r e a s e w i t h r e s p e c t t o b e n d i n g and t o r s i o n , The

i n f l u e n c e of b e n d i n g and t o r s i o n becomes f u l l y e f f e c t i v e o n l y

when t h e i m p r i n t s a r e e x t r e m e l y heavy o r where t h e r e i s c o r r o s i o n , C, A p p l i c a t i o n of t h e Formulae

The l i f e of s t r a n d e d s t e e l w i r e r o p e s c a n be d e t e r m i n e d from t h e number of bends t o f a i l u r e

-

t h e o p e r a t i n g , con- d i t i o n s , of c o u r s e , p l a y a n i m p o r t a n t r o l e . Comparison

t e s t s on t h e l i f e of w i r e r o p e s a l w a y s r e q u i r e i d e n t i c a l o p e r a t i n g c o n d i t i o n s , To c h a r a c t e r i z e t h e number o f b e n d s t o f a i l u r e , nB, f o r w i r e r o p e s of a c e r t a i n t y p e a r e l a t i o n b a s e d on a d e s i g n c r i t e r i o n g i v e n by D r u c k e r and Tachau ( 5 ) i s i n t r o d u c e d , v i z , , B = S _{= c o n s t a n t f o r a c e r t a i n nB}

,

~ D P

z where S = t e n s i o n i n w i r e r o p e , d = d i a m e t e r of w i r e r o p e , D = p i t c h d i a m e t e r o f s h e a v e ,

4

= t e n s i l e s t r e n g t h o f t h e w i r e s B = a c o m p a r a t i v e v a l u e w h i c h i s c o n s t a n t f o r c e r t a i n nB and c e r t a i n t y p e of r o p e . The s t r e s s e s a r e assumed t o be a s d e f i n e d by The v a l u e s l i s t e d i n T a b l e V were e x p e r i m e n t a l l y d e t e r - mined b y D r u c k e r and Tachau f o r r e g u l a r l a y c r a n e r o p e s ,

t y p e B, o f 6 x 37 c o n s t r u c t i o n , For number o f bends t o 6

f a i l u r e n = 1 x 1 0

,

t e n s i o n S = 2,500 kgm,, t e n s i l e s t r e n g t h o f w i r e p z = 1 6 0 kgm0 p e r s q . mm.,

dD C 1 9 , 5 0 0 = c o n s t a n t , hence, D % 19,50O/d

.

T a b l e V -2 ) o b t a i n e d by e x t r a p o l a t i o n . S i n c e f o r a w i r e r o p e t y p e B t h e r a t i o d/6 i s a p p r o x - i m a t e l y 22, d/& Z 8 8 5 = c o n s t a n t , ( 3 5 ~ ) By means o f t h i s r e l a t i o n t h e v a l u e s r e q u i r e d Tor D c a n be d e t e r m i n e d f o r t h e v a r i o u s r o p e s o f t y p e B, i f

8

o r d i s g i v e n .

The f o r m u l a e d e r i v e d f o r s e c o n d a r y b e n d i n k and com- p r e s s i o n s w i l l now be a p p l i e d t o t h e s e w i r e r o p e s , a s s u m i n k t h a t t h e y a r e r e g u l a r l a y r o p e s w i t h t h e f o l l o w i n g d a t a : L e n g t h o f l a y o f t h e s t r a n d s i n w i r e r o p e

...

L = 7,5d, number of s t r a n d s ...,...,.o..,..~D.~O.Oee..., z = 6, t e n s i o n i n t h e r o p e

....,,.,o...o...

S = 2500 k p . ; l e n k t h o f l a y o f a w i r e i n a s t r a n d

...

L'

= 10da

.

number o f o u t e r m o s t w i r e s

...,....,,..,..,...,...

z f = 18, t e n s i l e s t r e n g t h of t h e wire. 0 * o 0 u o o e o o o 0 o o o o e o o a 6 Je

=

1 6 0 kgmo p e r sq. mm,

( F o r c o r r e s p o n d i n g s t r a n d c o n s i s t i n g of 37 w i r e s , s e e F i g , 3.) Furthermore, i t i s assumed t h a t t h e sheave i s of g r a y c a s t i r o n and t h a t t h e groove i s t i g h t .

( a ) D e t e r m i n a t i o n o f t h e compressive f o r c e s Po

( i ) Between w i r e rope and sheave, a c c o r d i n g t o e q u a t i o n ( 4 4 , P o -

- -

z d D

6

-

4L

'

S = 12,500 kgrn. ( i i ) Between two s t r a n d s , a c c o r d i n g t o e q u a t i o n (15), where 0

Y

= 60

,

a = 15O30f, dl= 15O. ( b ) D e t e r m i n a t i o n of t h e compressions

( I ) Between w i r e r o p e and sheave

If El = 12,000 kgrn. p e r s q ,

mm.

and E2 = 20,000 kgm. p e r s q . m, a r e t a k e n a s t h e v a l u e s f o r g r a y c a s t i r o n and s t e e l r e s p e c t i v e l y , t h e n E = 15,000 kgm. p e r s q . mm, T h e r e f o r e , by e q u a t i o n s ( 1 7 ) and ( 1 8 a ) ,

m = 1.05d =

-

1.05 o 216,

-

22; = D/B

.

6

6

Thus * 1

B

I

6

D - 0 . 9 4 8 3 - D c o s

cr

=

-

(40)

According t o F i g .

l l p v

and po a r e o b t a i n e d from c o s

T

by s u b s t i t u t i o n of Po,

M

a n d / r v i n e q u a t i o n ( 3 8 ) . Here

I

0.966, c o s

r 4

0.978,

pv

1 . 8 5 and ( i i ) Between two s t r a n d s Assuming t h a t El = E2 = 20,000 kgm. p e r s q .

rmn.

and, t h e r e f o r e , c o r r e s p o n d i n g t o e q u a t i o n ( 2 0 a ) , _- hioreover, by e q u a t i o n s

( 1 8 ~ 1 ,

( 2 2 ) and ( 2 3 ) , t h u s hl = 2 i l

+

= 2.0224

(1 -*I

C O S T

=

-

c o s 30' = 0,845 T h e r e f o r e , a c c o r d i n g t o F i g .

11,/~lr

= 1.3, whence = -280 Po = -215 1 , 3 i . e , , i n a n a l o g y w i t h e q u a t i o n ( 2 4 ) and

Fig,

1 2 , ( c ) D e t e r m i n a t i o n of t h e comparison s t r e s s e s

( i ) P o i n t s o f c o n t a c t between w i r e r o p e and sheave

It may be assunled t h a t t h e r a t i o of t h e p r i n c i p a l a x e s of a n e l l i p t i c a l i m p r i n t , a / b 9 e q u a l s 1 0 . T h e r e f o r e , t h e f o l l o w i n g i s o b t a i n e d by e q u a t i o n s ( 2 6 ) , ( 2 8 a ) and ( 2 8 d ) : C 0 . 3 5 po +

(6,

- c b ) s where i t i s assumed t h a t _{d b}

_=cbl

( i i ) P o i n t s of c o n t a c t between two s t r a n d s

I n t h i s c a s e a/b 3 and, by e q u a t i o n s ( 2 8 ) , ( 2 8 a ) and

S i n c e t h e a n g l e between t h e a x i s of b e n d i n g and t h e d i r e c t i o n 0

of p r e s s u r e i s 30

,

cb

= 0.5

( d ) C a l c u l a t i o n o f t h e s t r e s s due t o pr3imary b e n d i n g , 1 T h i s s t r e s s i s c a l c u l a t e d f r o m t h e c o n v e n t i o n a l e q u a t i o n assuming t h a t E = 20,000 kgm, p e r sq.. rnm, ( e ) D e t e r m i n a t i o n o f t h e s t r e s s e s due t o s e c o n d a r y bendink: I f t h e i n t e r s e c t i n g a n g l e LJ between t h e two o u t e r m o s t w i r e s o f a s t r a n d h a v i n g 37 i d e n t i c a l wires i s 30°, t h e n , .by e q u a t i o n ( 6 b ) , If w i s s m a l l e r t h a n 3009

a

i n c r e a s e s , i . e . , i t becomes b2 more u n f a v o u r a b l e , ( f ) D e t e r m i n a t i o n o f she s t r e s s Gadmissible f o r a l o a d of one m i l l i o n c y c l e s , E x p e r i m e n t s c a r r i e d o u t by t h e EMPA ( c f . T a b l e V I I I ) showed t h a t w i t h 1 , 5 f o l d s a f e t y a g a i n s t f a i l u r e f o r v i i r e s h a v i n g a s t r e n g t h of 160 kgm. p e r s q , mix, I t i s assumed h e r e t h a t t h e w i r e r o p e i s always b e n t i n t h e same d i r e c t i o n ,

The r e s u l t s o b t a i n e d a r e compiled i n T a b l e V I and may be summarized a s f o l l o w s :

1. Dimensions

d i a m e t e r of a s i n g l e w i r e ,

8

,

t h e d i a m e t e r of sheave, D, and t h e c r o s s - s e c t i o n of w i r e rope, f . The v a l u e s f o r D were c a l c u l a t e d from e q u a t i o n ( 3 5 a ) , where B = 0.0008. Furthermore, i t i s assumed t h a t f o r a l l t h e w i r e r o p e s t h e l e n k t h of l a y of t h e s t r a n d s i s L = 7.5d, and t h a t of t h e w i r e s

Ll

= l o d e I t i s a l s o assumed t h a t t h e i n t e r s e c t i n g a n g l e between t h e two outermost w i r e s of a s t r a n d i s

a =

30' and t h a t t h e g r a y c a s t i r o n s h e a v e s have t i g h t grooves.

2. Compressions

The t a b l e g i v e s t h e compressive f o r c e s , Po, t h e maxi- mum compressions, po, and t h e comparison s t r e s s ,

u

g o

( a ) P o i n t s of c o n t a c t between sheave and rope

If i t i s assumed t h a t B = c o n s t a n t f o r a l l t h e wire' r o p e s of t y p e B, t h e n t h e maximum compressions po between sheave and w i r e rope a r e a l s o p r a c t i c a l l y c o n s t a n t . These v a l u e s a r e s m a l l e r than 2/, t h r o u g h o u t . Moreover, w i t h one e x c e p t i o n , a l l t h e 6 a r e s m a l l e r t h a n 0 . 7 5 ~ ~ . g ( b ) P o i n t s of c o n t a c t between two s t r a n d s For t h e s e p o i n t s o f c o n t a c t t h e v a l u e s f o r po and a r e c o n s i d e r a b l y h i g h e r . T h i s i n d i c a t e s e a r l y f o r m a t i o n of permanent i m p r i n t s a t t h e p o i n t s of c o n t a c t between two s t r a n d s , t h e dimensions depending on t h e v a l u e pmean

-

230 kgm. p e r s q . mm. ( F i g . 1 5 ) ,

3 , S t r A e s s e s The f o l l o w i n g s L r a e s s e s arc; l i s t e d i n t h e t a b l e :

r z ,

and b l

*

b2' a s t h e y o c c u r i n t h e t e n s i u r i zone or t h e w i r e s i n c o n t a c t w i t h t h e s h e a v e , and t h e i r c o m b i n a t i o n s a s w e l l a s (Tadma' i o e o , t h e maximum a d r r ~ i s s i b l s s t r e s s f o r one m i l l i o n c y c l e s , he s t r e s s (Tfailure i s 1.5 x cadmd

I t i s n o t e w o r t h y t h a t t h e s t r e s s due t o s e c o n d a r y b e n u i n k ,

eb2'

i s p r a c t i c a l l y e q u a l l o r a l l w i r e raopes of t y p e

B.

With one e x c e p t i o n , t h e s t r e s s e s = % + C b

max , 1 + (Tbg

a r e lower t h a n t h e v a l u e 6 _adm, and, t h e r e f o r e , a l s o l o w e r t h a n t h e v a l u e

ufailure0

4 , Summary

The f o l l o w i n g c o n c l u s i o n i s drawn from Drucker and Tachauqs s t a t e m e n t t h a t t h e l i f e o f w i r e r o p e s of a c e r t a i n t y p e may b e e x p r e s s e d by a c o n s t a n t v a l u e B: The maxir:~urn c o m p r e s s i o n s between s h e a v e and r o p e a s w e l l a s t h e s t r e s s e s due t o s e c o n d a r y bending, Cf

b2 a r e a l s o c o n s t a n t , prsovideds o f c o u r s e , t h a t t h e a n g l e s of l a y a l w a y s r e m a i n corlstant;, Moreover, f o r t h e l o a d f a c t o r nE = 1 X 1 ~ ' s t r e s s v a l u e s a r e

o b t a i n e d which, takink; t h e v a l u e s of Ofb i n t o a c c o u n t ,

2

v i r t u a l l y a g r e e w i t h t h e v a l u e s e x p e r i m e n t a l l y o b t a i n e a f o r %dm, by t h e EMPA. With t h e u s e o f t h e c o r r e s p o n d i n g

v a l u e B it s h o u l d b e p o s s i b l e t o o b t a i n a n a l o b o u s v a l u e s

f o r o t h e r t y p e s o f w i r e r o p e a s w e l l ,